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Abstract
In this paper, we introduce a cofibrant simplicial category that we call the free homotopy coherent adjunction and characterise its narrows using a graphical calculus that we develop here. The homspaces are appropriately fibrant, indeed are nerves of categories, which indicates that all of the expected coherence equations in each dimension are present. To justify our terminology, we prove that any adjunction of quasicategories extends to a homotopy coherent adjunction and furthermore that these extensions are homotopically unique in the sense that the relevant spaces of extensions are contractible Kan complexes. We extract several simplicial functors from the free homotopy coherent adjunction and show that quasicategories are closed under weighted limits with these weights. These weighted limits are used to define the homotopy coherent monadic adjunction associated to a homotopy coherent monad. We show that each vertex in the quasicategory of algebras for a homotopy coherent monad is a codescent object of a canonical diagram of free algebras. To conclude, we prove the quasicategorical monadicity theorem, describing conditions under which the canonical comparison functor from a homotopy coherent adjunction to the associated monadic adjunction is an equivalence of quasicategories. Our proofs reveal that a mild variant of Beck's argument is "all in the weights"much of it independent of the quasicategorical context.
Original language  English 

Pages (fromto)  802888 
Number of pages  87 
Journal  Advances in Mathematics 
Volume  286 
DOIs  
Publication status  Published  2 Jan 2016 
Keywords
 Adjunction
 Homotopy coherence
 Monad
 Monadicity
 Quasicategories
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Dive into the research topics of 'Homotopy coherent adjunctions and the formal theory of monads'. Together they form a unique fingerprint.Projects
 1 Finished

Applicable categorical structures
Street, R., Johnson, M., Lack, S., Verity, D. & Lan, R.
1/01/10 → 30/06/14
Project: Research